Calculus Examples

$lim_{x \to 3} \arctan (\frac{x + 3}{3 x})$

Step 1

Simplify the answer.

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Step 1.1

Cancel the common factor of $3 + 3$ and $3$ .

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Step 1.1.1

Factor $3$ out of $3$ .

$\frac{1}{3} \cdot \frac{3 \cdot 1 + 3}{3}$

Step 1.1.2

Factor $3$ out of $3$ .

$\frac{1}{3} \cdot \frac{3 \cdot 1 + 3 \cdot 1}{3}$

Step 1.1.3

Factor $3$ out of $3 \cdot 1 + 3 \cdot 1$ .

$\frac{1}{3} \cdot \frac{3 \cdot (1 + 1)}{3}$

Step 1.1.4

Cancel the common factors.

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Step 1.1.4.1

Factor $3$ out of $3$ .

$\frac{1}{3} \cdot \frac{3 \cdot (1 + 1)}{3 (1)}$

Step 1.1.4.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{3 \cdot (1 + 1)}{3 \cdot 1}$

Step 1.1.4.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{1 + 1}{1}$

Step 1.1.4.4

Divide $1 + 1$ by $1$ .

$\frac{1}{3} (1 + 1)$

Step 1.2

Add $1$ and $1$ .

$\frac{1}{3} \cdot 2$

Step 1.3

Combine $\frac{1}{3}$ and $2$ .

$\frac{2}{3}$

Step 2

Substitute $t$ for $\frac{x + 3}{3 x}$ and let $t$ approach $\frac{2}{3}$ since $lim_{x \to 3} \frac{x + 3}{3 x} = \frac{2}{3}$ .

$lim_{t \to \frac{2}{3}} \arctan (t)$

Step 3

Evaluate the limits by plugging in $\frac{2}{3}$ for all occurrences of $t$ .

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Step 3.1

Simplify the answer.

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Step 3.1.1

Cancel the common factor of $3 + 3$ and $3$ .

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Step 3.1.1.1

Factor $3$ out of $3$ .

$\arctan (\frac{1}{3} \cdot \frac{3 \cdot 1 + 3}{3})$

Step 3.1.1.2

Factor $3$ out of $3$ .

$\arctan (\frac{1}{3} \cdot \frac{3 \cdot 1 + 3 \cdot 1}{3})$

Step 3.1.1.3

Factor $3$ out of $3 \cdot 1 + 3 \cdot 1$ .

$\arctan (\frac{1}{3} \cdot \frac{3 \cdot (1 + 1)}{3})$

Step 3.1.1.4

Cancel the common factors.

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Step 3.1.1.4.1

Factor $3$ out of $3$ .

$\arctan (\frac{1}{3} \cdot \frac{3 \cdot (1 + 1)}{3 (1)})$

Step 3.1.1.4.2

Cancel the common factor.

$\arctan (\frac{1}{3} \cdot \frac{3 \cdot (1 + 1)}{3 \cdot 1})$

Step 3.1.1.4.3

Rewrite the expression.

$\arctan (\frac{1}{3} \cdot \frac{1 + 1}{1})$

Step 3.1.1.4.4

Divide $1 + 1$ by $1$ .

$\arctan (\frac{1}{3} (1 + 1))$

Step 3.1.2

Add $1$ and $1$ .

$\arctan (\frac{1}{3} \cdot 2)$

Step 3.1.3

Combine $\frac{1}{3}$ and $2$ .

$\arctan (\frac{2}{3})$

Step 3.2

Evaluate $\arctan (\frac{2}{3})$ .

$0.5880026$